Probability

Discrete Random Variables

Discrete Random Variables - Joint Probability Distribution

         

The joint probability distribution of discrete random variables X X and Y Y is given as follows: P(x,y)x=0x=1x=2y=00.320.160.32y=10.080.040.08 \begin{matrix} P(x,y) & x=0 & x=1 & x=2 \\ y=0 & 0.32 & 0.16 & 0.32 \\ y=1 & 0.08 & 0.04 & 0.08 \end{matrix} Are X X and YY independent of each other?

Today, Joe's class took a math quiz consisting of two problems. For a randomly selected student, let X X be the points earned on the first question and Y Y be the points earned on the second question. The joint probability distribution of X X and Y Y is given in the following table:

X=0X=5X=10X=15Y=00.050.0500.07Y=5000.180.07Y=1000.110.360.11 \begin{matrix} & X=0 & X=5 & X=10 & X=15 \\ Y=0 & 0.05 & 0.05 & 0 & 0.07 \\ Y=5 & 0 & 0 & 0.18 & 0.07 \\ Y=10 & 0 & 0.11 & 0.36 & 0.11 \end{matrix}

What is the expected points that a randomly selected student got?

If the joint probability distribution of X X and Y Y is given by as follows: P(x,y)x=0x=1x=2x=3y=000.0500.04y=10.040.410.050.05y=20.050.040.040.05y=30.050.040.050.04 \begin{matrix} P(x,y) & x=0 & x=1 & x=2 & x=3 \\ y=0 & 0 & 0.05 & 0 & 0.04 \\ y=1 & 0.04 & 0.41 & 0.05 & 0.05 \\ y=2 & 0.05 & 0.04 & 0.04 & 0.05 \\ y=3 & 0.05 & 0.04 & 0.05 & 0.04 \end{matrix} what is E[X+Y]? E[X+Y]?

If the joint probability P(x,y)P(x,y) of X X and Y Y is given by as follows: P(x,y)x=0x=1x=2x=3x=4y=000.030.030.050.04y=10.2900.060.050.04y=200.060.030.050.05y=30.050.050.050.030.04 \begin{matrix} P(x,y) & x=0 & x=1 & x=2 & x=3 & x=4 \\ y=0 & 0 & 0.03 & 0.03 & 0.05 & 0.04 \\ y=1 & 0.29 & 0 & 0.06 & 0.05 & 0.04 \\ y=2 & 0 & 0.06 & 0.03 & 0.05 & 0.05 \\ y=3 & 0.05 & 0.05 & 0.05 & 0.03 & 0.04 \end{matrix} what is E[XY]? E[XY]?

Today, Joe's class took a math quiz consisting of two problems. For a randomly selected student, let X X be the points earned on the first question and Y Y be the points earned on the second question. The joint probability distribution of X X and Y Y is given in the following table:

X=0X=5X=10X=15Y=00.070.0700.07Y=5000.180.07Y=1000.110.320.11 \begin{matrix} & X=0 & X=5 & X=10 & X=15 \\ Y=0 & 0.07 & 0.07 & 0 & 0.07 \\ Y=5 & 0 & 0 & 0.18 & 0.07 \\ Y=10 & 0 & 0.11 & 0.32 & 0.11 \end{matrix}

The students who have earned a total of less than 1515 points will be taking a makeup test. What is the probability that a randomly chosen student will be taking a makeup test?

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